Cardiac memory phenomenon, time-fractional order nonlinear system and bidomain-torso type model in electrocardiology

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nonlinear system and bidomain-torso type model in electrocardiology

Aziz Belmiloudi

To cite this version:

Aziz Belmiloudi. Cardiac memory phenomenon, time-fractional order nonlinear system and bidomain- torso type model in electrocardiology. AIMS Mathematics, AIMS Press, 2021, 6 (1), pp.821-867.

�10.3934/math.2021050�. �hal-03093927�

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http: // www.aimspress.com / journal / Math

DOI:10.3934 / math.2021050 Received: 11 June 2020 Accepted: 11 October 2020 Published: 02 November 2020

Research article

Cardiac memory phenomenon, time-fractional order nonlinear system and bidomain-torso type model in electrocardiology

Aziz Belmiloudi

^∗

Institut de Recherche Mathématique de Rennes (IRMAR), Université Européenne de Bretagne, 20 avenue Buttes de Coësmes, CS 70839, 35708 Rennes Cédex 7, France

* Correspondence: Email: aziz.belmiloudi@math.cnrs.fr.

Abstract: Cardiac memory, also known as the Chatterjee phenomenon, refers to the persistent but reversible T-wave changes on ECG caused by an abnormal electrical activation pattern. After a period of abnormal ventricular activation in which the myocardial repolarization is altered and delayed (such as with artificial pacemakers, tachyarrhythmias with wide QRS complexes or ventricular pre-excitation), the heart remembers and mirrors its repolarization in the direction of the vector of

“abnormally” activated QRS complexes. This phenomenon alters patterns of gap junction distribution and generates changes in repolarization seen at the level of ionic-channel, ionic concentrations, ionic- current gating and action potential. In this work, we propose a mathematical model of cardiac electrophysiology which takes into account cardiac memory phenomena. The electrical activity in heart through torso, which is dependent on the prior history of accrued heartbeats, is mathematically modeled by a modified bidomain system with time fractional-order dynamics (which are used to describe processes that exhibit memory). This new bidomain system, that I name “ memory bidomain system”, is a degenerate nonlinear coupled system of reaction-di ff usion equations in shape of a fractional- order di ff erential equation coupled with a set of time fractional-order partial di ff erential equations.

Cardiac memory is represented via fractional-order capacitor (associate to capacitive current) and fractional-order cellular membrane dynamics. First, mathematical model is introduced. Afterward, results on generalized Gronwall inequality within the framework of coupled fractional di ff erential equations are developed. Next, the existence and uniqueness of solution of state system are proved as well as stability result. Further, some preliminary numerical applications are performed to show that memory reproduced by fractional-order derivatives can play a significant role on key dependent electrical properties including APD, action potential morphology and spontaneous activity.

Keywords: fractional-order dynamics; heart-torso coupling; cellular heterogeneity; integral inequality; weak solution; cardiac memory; memory induced T-wave; ionic models

Mathematics Subject Classification: 35R11, 34A08, 35K57, 35B30, 92C50, 35Q92

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1. Introduction and mathematical setting of the problem 1.1. Modeling motivation

The heart is an electrically controlled mechanical pump which drives blood flow through the circulatory system vessels (through deformation of its walls), where electrical impulses trigger mechanical contraction (of various chambers of heart) and whose dysfunction is incompatible with life. The coordinated contraction of heart and the maintenance of heartbeat are controlled by a complex network of interconnected cardiac cells electrically coupled by gap junctions and voltage-gated ion channels. The evaluation of bioelectrical activity in heart is then a very complex process which uses di ff erent phenomenological mechanism and subject to various perturbations, and physiological and pathophysiological variations.

The electrical system of a normal heart is highly organized in a steady rhythmic pattern. This normal heartbeat is called sinus rhythm. Irregular or abnormal heartbeats, called arrhythmias, are caused by a change in propagation and / or formation of electrical impulses, that regulate a steady heartbeat, causing a heartbeat that is too fast or too slow, that can remain stable or become chaotic (irregular and disorganized). Many times, arrhythmias are harmless and can occur in healthy people without heart disease; however, some of these rhythms can be serious and require special and e ffi ciency treatments.

Fibrillation is one type of arrhythmia and is considered the most serious cardiac rhythm disturbance. It occurs when the heart beats with rapid, erratic electrical impulses (highly disorganized almost chaotic activation). This leads to quivering (or fibrillation) of heart chambers rather than normal contraction, which then leads the heart to lose its ability to pump enough blood through circulatory system. The treatment therapy of these diseases, when it becomes troublesome or when it can present a danger, often uses electrical impulses to stabilize cardiac function and restore the sinus rhythm, by implanting the patients with active cardiac devices (electrotherapy). For example, in case of cardiac rhythms that are too slow, the devices transmit electronic impulses and ensure that periodic contractions of heart are maintained at a hemodynamically su ffi cient rate; and in the case of a fast or irregular heart rate, the devices monitor heart rate and, if needed, treat episodes of tachyarrhythmia (including tachycardia and / or fibrillation) by transmitting automatically impulses to either give defibrillation shocks or cause overstimulation (via an ICDs

^∗

) or synchronize contraction of left and right ventricles.

After cessation of a transient period of abnormal ventricular activation (arrhythmia or pacing) in which the myocardial repolarization is altered and delayed (such as with artificial pacemakers [82], tachyarrhythmias with wide QRS complexes, intermittent left bundle branch block or ventricular pre-excitation observed in Wol ff -Parkinson-White syndrome [45]), the heart remembers and mirrors its repolarization in the direction of vector of “abnormally” activated QRS complexes [66]. This remodeling of electrical properties of myocardium is characterized by persistent but reversible T-wave changes on the surface electrocardiogram (ECG). The scope, significance and direction of T-wave deviation depend on duration and direction of abnormal electrical activation. Moreover, these changes are often confused with pathological conditions manifesting with T-wave deviations, such as (acute) myocardial ischemia or infarction. This cumulative and complex phenomenon is named cardiac memory and can persist up for several weeks after normal ventricular conduction is restored. Heart is considered as network of cardiac oscillators communicating via gap junctions between neighboring cells and through voltage gated ion channels (that are activated by changes in electrical membrane

∗The so-called implantable cardioverter defibrillators

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potential near channel): this phenomenon alters patterns of gap junction distribution and generates changes in repolarization seen at the level of ionic-channel, ionic concentrations, ionic-current gating and action potential.

The existence of cardiac memory has been known for many years and resulted in a large number of publications, see for example [23, 37, 44, 58–60, 67, 72, 73, 75] and the references therein. Yet despite all this, this phenomemon is under-recognized and there is still limited information regarding its physiological significance and practical clinical implications because this phenomenon is considered to be a relatively benign pathophysiologic finding. Unfortunately, the work of past few years has shown that despite the mostly benign nature of cardiac memory in healthy individuals under some conditions, it can be the trigger of more complex arrhythmias and then requires emergency care of the patient. Other works estimate that clinically administered antiarrhythmic drugs alter expression of cardiac memory and that generate changes in repolarization could in turn alter the e ff ects of these drugs (see e.g. [4, 63]). Moreover, cardiac memory may also lead to unnecessary and invasive diagnostic investigation that put patients under unnecessary risks (see e.g., [71]).

Then, recognition of cardiac memory as a serious potential cause of T-wave changes and the analysis of T-wave morphology (throughout the ECG during narrow- and wide-QRS rhythms) are critical to help di ff erentiate T-wave changes due to myocardial ischemia from those induced by cardiac memory, and consequently sustainably establish the clinical relevance of this phenomenon, facilitates diagnosis and increases e ffi ciency of cardiac disorders treatment.

Consequently, this has greatly emphasized the need for model and methodologies capable of predicting and understanding the dynamic mechanisms of di ff erent sources of electrical instability in heart like cardiac action potential (AP) repolarization alternans which is influenced by memory. At cellular level, alternans is generally manifests as cyclic, beat-to-beat alternations between long and short action potential and / or intracellular Ca transient, and is frequently associated with the development of ventricular tachycardia and fibrillation. It is generally agreed that disturbances of bi-directional (mutual) relationship between transmembrane potential (or membrane voltage) φ and Ca-sensitive ionic currents play a key role in generation of alternans. Because membrane voltage φ is strongly a ff ected by Ca-sensitive ionic currents and intracellular Ca loading in turn is strongly influenced by φ-dependent ionic currents. This complex bidirectional coupling influences and controls the amplitude and duration of action potentials (APD) through various time- and voltage-dependent ionic currents.

The classical bidomain system is commonly used for modeling propagation of electrophysiological waves in cardiac tissue. Motivated by above discussions, to take into account the critical e ff ect of memory in propagation of electrophysiological waves, together with other critical cardiac material parameters, we propose and analyze a new bidomain model, that I name “memory bidomain system”

by incorporating memory e ff ects. In next section, we shall present derivation of this memory bidomain model.

1.2. Modeling and formulation of the problem

Mathematical and computational cardiac electrophysiological modeling is now an important field

in applied mathematics. Indeed, nowadays, heart and cardiovascular diseases are still the leading cause

of death and disability all over the world. That is why we need to improve our knowledge about heart

behavior, and more particularly about its electrical behavior. Consequently we want strong methods to

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compute electrical fluctuations in the myocardium to prevent cardiac disorders (as arrhythmia). Tissue- level cardiac electrophysiology, which can provide a bridge between electrophysiological cell models at smaller scales, and tissue mechanics, metabolism and blood flow at larger scales, is usually modeled using the coupled bidomain equations, originally derived in [78], which represent a homogenization of the intracellular and extracellular medium, where electrical currents are governed by Ohm’s law (see also e.g. [46] for a review and an introduction to this field). The model was modified and extended to include heart tissue surrounded by a conductive bath or a conductive body (see e.g. [64] and [74]).

From mathematical viewpoint, the classical bidomain system is commonly formulated in terms of intracellular and extracellular electrical potentials of anisotropic cardiac tissue (macroscale), φ

i

and φ

e

, (or, equivalently, extracellular potential φ

e

and the transmembrane voltage φ = ϕ

i

− ϕ

e

) coupled with cellular state variables u describing cellular membrane dynamics and torso potential state variable φ

s

. This is a system of non-linear partial di ff erential equations (PDEs) coupled with ordinary di ff erential equations (ODEs), in the physical region Ω (occupied by excitable cardiac tissue, which is an open, bounded, and connected subset of d-dimensional Euclidean space R

^d

, d ≤ 3). The PDEs describe the propagation of electrical potentials and ODEs describe the electrochemical processes.

Cardiac memory can a ff ect considerably the resulting electrical activity in heart and thus the cardiac disorders therapeutic treatment. It is then necessary to introduce the impact of memory on dynamical behaviors of such a system. Memory terms can cause dynamical instabilities (as Alternans) and give rise to highly complex behavior including oscillations and chaos.

Figure 1. The derived “Memory bidomain model” is defined on heart domain Ω

H

, while Ω

B

is the rest of body.

In order to take into account the influence of cardiac memory and inward movement of u into the cell which prolongs depolarization phase of action potential, we propose a new bidomain model.

In this new model, classical bidomain model has been modified via time fractional-order dynamical system, arising due to cellular heterogeneity, which are used to describe processes that exhibit memory.

More precisely, the derived system with memory (or history), is a nonlinear coupled reaction-di ff usion

model in shape of a set of time fractional-order di ff erential equations (FDE) coupled with a set of time

fractional-order partial di ff erential equations, in the torso-heart’s spatial domain Ω (Figure 1) which is

a bounded open subset with a su ffi ciently regular boundary ∂ Ω , and during the final fixed time horizon

T > 0, as follows

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c

_α

∂

^α₀+

φ + I(.; φ, u) − div(K

i

∇ϕ

i

) = I

i

, in Q

_H

= Ω

H

× (0, T ) c

_α

∂

^α₀+

φ + I(.; φ, u) + div(K

e

∇ϕ

e

) = −I

e

, in Q

_H

−div(K

s

∇ϕ

s

) = 0, in Q

_B

= Ω

B

× (0, T )

∂

^β₀₊

u + G(.; φ, u) = 0, in Q

_H

subject to initial and boundary conditions (1.4),

(1.1)

or equivalently

c

_α

∂

^α₀+

φ + I(.; φ, u) − div(K

i

∇φ) = div(K

i

∇ϕ

e

) + I

i

, in Q

_H

−div((K

e

+ K

_i

)∇ϕ

e

) = div(K

i

∇φ) + I, in Q

_H

−div(K

s

∇ϕ

s

) = 0, in Q

_B

∂

^β₀₊

u + G(.; φ, u) = 0, in Q

_H

subject to initial and boundary conditions (1.4).

(1.2)

Here, ∂

^α₀₊

and ∂

^β₀₊

denote the forward Caputo fractional derivatives with α and β be real values in ]0, 1] and the unknowns are the potentials ϕ

i

, ϕ

e

, ϕ

s

and a single ionic variable u (e.g. gating variable, concentration, etc.).

The heart’s spatial domain is represented by Ω

H

which is a bounded open subset, and by Γ

H

=

∂ Ω

H

we denote its piecewise smooth boundary. A distinction is made between the intracellular and extracellular tissues which are separated by the cardiac cellular membrane. The surrounding tissue within the thorax is modeled by a volume conduction Ω

B

with a piecewise smooth boundary Γ

B

= Γ

H

∪ Γ

T

where Γ

T

is the thorax surface. The whole domain is denoted by the Ω = Ω

H

∪ Ω

B

. In Ω

H

, the transmembrane potential is φ = ϕ

i

− ϕ

e

where ϕ

e

and ϕ

i

are the transmembrane, extracellular and intracellular potentials, respectively, and in Ω

B

, ϕ

s

is thoracic medium electric potential. The parameter c

_α

is c

_α

= κC

α

> 0, where C

α

is the membrane capacitance per unit area and κ is the surface area-to- volume ratio (homogenization parameter). The membrane is assumed to be passive, so the capacitance C

_α

can be assumed to be not a function of state variables. Classically this membrane is assimilate to a simple parallel resistor-capacitor circuit. However various studies showed clearly that a passive membrane may be more appropriately modeled with a non-ideal capacitor, in which the current-voltage relationship is given by a fractional-order derivative (see e.g [84]). So, according to the passivity of tissue we can assimilate this membrane to electrical circuit with a resistor associate to the ionic current (I

ion

) and a capacitor associate to the capacitive current, I

_c^α

= c

_α

∂

^α₀₊

φ, in parallel, with α usually ranging from 0.5 to 1 (Figure 2). Moreover, since the electrical restitution curve (ERC)

^†

is a ff ected by the action potential history through ionic memory, we have represented the memory via ionic variable u by a time fractional-order dynamic term ∂

^β₀₊

u.

†which traditionally describes the recovery of APD as a function of the interbeat interval

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Figure 2. Modeling of the membrane as resistor and non-ideal capacitor coupled in parallel (where 0 < α ≤ 1 is the fractional order of capacitor).

The coupling between equations in Ω

H

and equation in Ω

B

(in systems (1.1) and (1.2)) is operate at the heart-torso interface Γ

H

. The continuity conditions at the interface can be given by

ϕ

e

= ϕ

s

, in Σ

H

= Γ

H

× (0, T )

K

_e

∇ϕ

e

· n = K

_s

∇ϕ

s

· n, on Σ

H

. (1.3) where n is the outward normal to Γ

H

.

The tensors K

_i

and K

_e

are the conductivity tensors describing anisotropic intracellular and extracellular conductive media, and tensor K

_s

represents the conductivity tensor of thoracic medium.

The electrophysiological ionic state u describes a cumulative way of e ff ects of ion transport through cell membranes (which describes e.g., the dynamics of ion-channel and ion concentrations in di ff erent cellular compartments). The operator I is equal to κI

ion

, where the nonlinear operator I

ion

describes the sum of transmembrane ionic currents across cell membrane with u. The nonlinear operator G is representing the ionic activity in myocardium. Functional forms for I and G are determined by an electrophysiological cell model. The source terms are I

i

= κ f

i

, I

e

= κ f

e

and I = I

i

+ I

e

, where f

i

and f

e

describe intracellular and extracellular stimulation currents, respectively.

To close the system, we impose the following initial and boundary conditions initial conditions

φ(., t = 0) = φ

₀

, u(., t = 0) = u

0

, in Ω

H

and boundary conditions ϕ

e

= ϕ

s

, in Σ

H

K

_e

∇ ϕ

e

· n = K

_s

∇ ϕ

s

· n, on Σ

H

K

_i

∇ ϕ

i

· n = 0, on Σ

H

K

_s

∇ ϕ

s

· n

T

= 0, on Σ

^T

= Γ

^T

× (0, T )

(1.4)

where n

T

is the outward normal to Γ

^T

.

In absence of a grounded electrode, the bidomain equations are a naturally singular problem since

ϕ

e

and ϕ

s

, in system (1.2), only appear in the equations and boundary conditions through their

gradients. Moreover, the states ϕ

e

and ϕ

s

are only defined up to the same constant. Such problems

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have compatibility conditions determining whether there are any solution to the PDEs. This is easily found by integrating the second and third equations of (1.2) over the domain and using the divergence theorem with boundary conditions. Then the following conservation of the total current is derived (a.e. in (0, T ))

Z

ΩH

Idx = 0. (1.5)

Consequently, we must choose current I such that the compatibility condition (1.5) is satisfied. For the choose of intracellular stimulus, it is natural and usual (as is common) to take I

i

= −I

e

(denoted in the sequel by f

H

) i.e., setting the total current I to zero. It is clear that this condition does not correspond to zero extracellular stimulus, but that the extracellular stimulus current and the intracellular stimulus current are equal in magnitude and opposite in direction. Moreover, the functions ϕ

e

and ϕ

s

are defined within a class of equivalence, regardless of the same time-dependent function. This function can be fixed, for example, by setting the condition, (a.e. in (0, T))

Z

ΓH

ϕ

e

dx = Z

ΓH

ϕ

s

dx = p

H

where p

H

is a fixed time-dependent function. (1.6) Remark 1.1. 1. Condition (1.6) is used for pressure in oceanography (see e.g. [15]).

2. The functions K

_i

, K

_e

, H and G depend on the fiber extension ratio.

3. We can suppose, for example, that f

H

is only a time dependent source and is of the form f

H

(x, t) = θ(t)(χ

_Ω⁽¹⁾

H

(x) − χ

_Ω⁽²⁾

H

(x)), (1.7)

where χ

_Ω⁽ⁱ⁾

H

is the characteristic function of set Ω

⁽ⁱ⁾_H

, i = 1, 2. The support regions Ω

⁽¹⁾_H

and Ω

⁽²⁾_H

can be considered to represent an anode (positive electrode) and a cathode (negative electrode)

respectively.

In recent years, various problems concerning biological rhythmic phenomena and memory processes which can be included in a cardiac model in many ways (via delay model or time fractional dynamical model), have been studied. Via delayed system we can cited e.g., [11, 17, 20, 40, 41, 43, 69]

and the references therein. For problems associated with bidomain models with time-delay, the

literature is limited, to our knowledge, to [8, 9, 27]. In these references, in order to take into account

the influence of disturbance in data and the time-varying delays on propagation of

electrophysiological waves in heart, the authors have developed a new mathematical model and have

considered the theoretical analysis as well as numerical simulations (with real data) and validation of

developed model. Via time fractional dynamical model, the literature is also limited, we can cite

e.g. [26, 31]. In [26], the authors study, using a minimal cardiac cell model, the e ff ects of a

fractional-order time di ff usion for the voltage and for the ionic current gating. They have shown

numerically the interest of modeling memory through fractional-order and that with the model it is

able to analyze the influence of memory on some electrical properties as spontaneous activity and

alternans. Concerning problems associated with classical bidomain models various methods and

technique, as evolution variational inequalities approach, semi-group theory, Faedo-Galerkin method

and others, the studies of well-posedness of solutions have been derived in the literature (see

e.g., [12, 16, 18, 19, 24, 80] and the references therein); for development of multiscale mathematical

and computational modeling of bioelectrical activity in myocardial tissue and the formation of cardiac

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disorders (as arrhythmias), and their numerical simulations, which are based on methods as finite di ff erence method, finite element method or lattice Boltzmann method, have been receiving a significant amount of attention (see e.g., [5, 6, 21, 28–30, 35, 38, 46, 51, 65, 79, 81] and the references therein).

The rest of paper is organized as follows. In next section, we give some preliminaries results useful in the sequel. In Section 3, some preliminaries results concerning fractional calculus are given and results on generalized Gronwall inequality to a coupled fractional di ff erential equations are developed.

In Section 4 we shall prove the existence, stability and uniqueness of weak solutions of derived model, under some hypotheses for data and some regularity of nonlinear operators. Numerical experiments, using a modified Lattice Boltzmann Method for numerical simulations of the derived memory bidomain systems, are described in Section 5. In Section 6, conclusions are discussed.

2. Assumptions, notations and some fundamental inequalities

Let f ⊂ IR

^m

, m ≥ 1, be an open and bounded set with a smooth boundary and f

^T

= f × (0, T ). We use the standard notation for Sobolev spaces (see [1]), denoting the norm of W

^q,p

( f ) (q ∈ IN, p ∈ [1, ∞]) by k . k

_W^q,p

. In the special case p = 2, we use H

^q

( f ) instead of W

^q,2

( f ). The duality pairing of a Banach space X with its dual space X

⁰

is given by h., .i

X⁰,X

. For a Hilbert space Y the inner product is denoted by (., .)

Y

and the inner product in L

²

( Ω ) is denoted by (., .). For any pair of real numbers r, s ≥ 0, we introduce the Sobolev space H

^r,s

( f

^T

) defined by H

^r,s

( f

T

) = L

²

(0, T ; H

^r

( f )) ∩ H

^s

(0, T ; L

²

( f )), which is a Hilbert space normed by k v k

²

L²(0,T;H^r(f))

+ k v k

²

H^s(0,T;L²(f))

1/2

, where H

^s

(0, T ; L

²

( f )) denotes the Sobolev space of order s of functions defined on (0, T ) and taking values in L

²

( f ), and defined (see [52]) by H

^s

(0, T ; L

²

( f )) = [H

^q

(0, T, L

²

( f )), L

²

( f

^T

)]

_θ

, for s = (1 − θ)q with θ ∈ (0, 1) and q ∈ IN, and H

^q

(0, T ; L

²

( f )) = n

v ∈ L

²

( f

T

)|

^∂_∂t^j^vj

∈ L

²

( f

T

), for 1 ≤ j ≤ q o

. For a given Banach space X , with norm k.k

_X

, of functions integrable on f , we define its subspace X | IR = n

u ∈ X , Z

f

u = 0 o

that is a Banach space with norm k.k

_X

, and we denote by [u] the projection of u ∈ X on X | IR such that [u] = u − 1

mes( f ) Z

f

udx (with mes( f ) standing for Lebesgue measure of f ).

Lemma 2.1. (Poincar´e-Wirtinger inequality) Assume that 1 ≤ p ≤ ∞ and that f is a bounded connected open subset of IR

^d

with a su ffi ciently regular boundary ∂ Ω (e.g., a Lipschitz boundary).

Then there exists a Poincar´e constant C , depending only on Ω and p, such that for every function u in Sobolev space W

^1,p

( f ), we have

k [u] k

_Lp(f)

≤ C k∇ u k

_Lp(f)

.

Remark 2.1. From the Poincar´e-Wirtinger inequality, we can deduce that the H

¹

semi-norm and the

H

¹

norm are equivalent in the space H

¹

( f )| IR.

Remark 2.2. Let q be a nonnegative integer. We have the following results (see e.g. [1])

(i) H

^q

( f ) ⊂ L

^p

( f ), ∀ p ∈ [1,

_m−2q^2m

], with continuous embedding (with the exception that if 2q = m,

then p ∈ [1, + ∞[ and if 2q > m, then p ∈ [1, + ∞] ).

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(ii) (Gagliardo-Nirenberg inequalities) There exists C > 0 such that k v k

_Lp(f)

≤ C k v k

^θ_Hq(f)

k v k

^1−θ

L²(f)

, ∀v ∈ H

^q

( f ),

where 0 ≤ θ < 1 and p =

_m−2θq^2m

(with the exception that if q − m/2 is a nonnegative integer, then θ is

restricted to 0).

Finally, we introduce the spaces:

• H

H

= L

²

( Ω

H

), V

H

= H

¹

( Ω

H

), V = H

¹

( Ω ) (endowed with their usual norms) and U

H

= V

H

| IR,

• U

HB

= n

ψ ∈ H

¹

( Ω ) | Z

ΩH

ψdx = 0 o .

We will denote by V

⁰H

(resp. U

⁰H

) the dual of V

H

(resp. of U

H

). We have the following continuous embeddings, where p ≥ 2 if d = 2 and 2 ≤ p ≤ 6 if d = 3, p

⁰

is such that 1

p

⁰

+ 1 p = 1 V

H

⊂ H

H

⊂ V

⁰_H

, U

H

⊂ H

H

| IR ⊂ U

⁰_H

,

V

H

⊂ L

^p

( Ω

H

) ⊂ H

H

≡ ( H

H

)

⁰

⊂ L

^p⁰

( Ω

H

) ⊂ V

⁰_H

(2.1) and the injections V

H

⊂ H

H

and U

H

⊂ H

H

| IR are compact. We can now introduce the following spaces (where q > 1, p ≥ 2 and

¹_p

+

_p¹⁰

= 1)

D

^p

(0, T ) = L

^p

( Q

_H

) ∩ L

²

(0, T ; V

H

), and its dual D

⁰p

(0, T ) = L

^p⁰

( Q

_H

) + L

²

(0, T ; V

⁰H

) ⊂ L

^p⁰

(0, T ; V

⁰H

).

Remark 2.3. Space D

^p

(0, T ) is equipped with norm: k u k

_D_p

= max(k u k

_Lp(QH)

, k u k

_L2(0,T;VH)

) and its dual with norm: k v k

_D⁰_p

= inf

v=v1+v2

(k v

₁

k

_Lp0

(QH)

+ k v

₂

k

_L2(0,T;V⁰_H)

).

Definition 2.1. A real valued function H defined on D × IR

^q

, q ≥ 1, is a Carath´eodory function i ff H(.; v) is measurable for all v ∈ IR

^q

and H(y; .) is continuous for almost all y ∈ D.

Our study involves the following fundamental inequalities, which are repeated here for review:

(i) H¨older’s inequality:

Z

D

Π

i=1,k

f

i

dx ≤ Π

i=1,k

k f

i

k

_Lqi(D)

, where k f

i

k

_Lqi(D)

= Z

D

| f

i

|

^qⁱ

dx

!

1/qi

and X

1≤i≤k

1 q

i

= 1.

(ii) Young’s inequality (∀a, b > 0 and > 0): ab ≤

_p

a

^p

+

⁻_q^q/p

b

^q

, f or p, q ∈]1, + ∞[ and

¹_p

+

¹_q

= 1.

(iii) Minkowski’s integral inequality:

"Z

Ω

Z

t

0

| f (x, s) | ds

!

p

dx

#

^1/p

≤ Z

t

0

Z

Ω

| f (x, s) |

^p

dx

!

1/p

ds, for p ∈]1, + ∞[ and t > 0.

Finally, we denote by L(A; B) the set of linear and continuous operators from a vectorial space A into a vectorial space B, and by R

^∗

the adjoint operator to a linear operator R between Banach spaces.

From now on, we assume that the following assumptions hold for the nonlinear operators and tensor

functions appearing in our model.

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(H1) We assume that the conductivity tensor functions K

_θ

∈ W

^1,∞

( Ω

H

), θ ∈ {i, e} and K

_s

∈ W

^1,∞

( Ω

B

) are symmetric, positive definite matrix functions and that they are uniformly elliptic, i.e., there exist constants 0 < K

₁

< K

₂

such that (∀ψ ∈ IR

^d

)

K

₁

kψk

²

≤ ψ

^T

K

_θ

ψ ≤ K

₂

kψk

²

in Ω

H

,

K

₁

kψk

²

≤ ψ

^T

K

_s

ψ ≤ K

₂

kψk

²

in Ω

B

. (2.2) Remark 2.4. We can emphasize a specificity of tensors K

_e

and K

_i

(see e.g., [25]).

1. The tensors K

_e

(x) and K

_i

(x) have the same basis of eigenvectors Q(x) = (q

k

(x))

_1≤k≤d

in IR

^d

, which reflect the organization of muscle in fibers, and consequently K

i

(x) = Q(x) Λ

ⁱ

(x)Q(x)

^T

and K

_e

(x) = Q(x) Λ

e

(x)Q(x)

^T

, where Λ

i

(x) = diag((λ

_i,k

)

_1≤k≤d

) and Λ

e

(x) = diag((λ

_e,k

)

_1≤k≤d

).

2. The muscle fibers are tangent to Γ so that (for θ ∈ { i, e } ) : K

_θ

n = λ

θ,d

n, a.e., in Γ , with λ

θ,d

(x) ≥

λ > 0, λ a constant.

The operators I and G which describe electrophysiological behavior of the system can be taken as follows (a ffi ne functions with respect to u)

I(x, t; φ, u) = I

₀

(x, t; φ) + I

₁

(x, t; φ)u,

G(x, t; φ, u) = I

₂

(x, t; φ) + ~ (x, t)u, (2.3) where ~ is a su ffi ciently regular function and I

₀

(x, t; φ) = g

₁

(x; φ) + g

₂

(x, t; φ) with g

₁

is an increasing function on φ. Moreover, the operators I

₀

, I

₁

and I

₂

appearing in I and G , are supposed to satisfy the following assumptions.

(H2)

_p

The operators I

₀

, I

₁

and I

₂

are Carath´eodory functions from ( Ω × IR) × IR into IR and continuous on φ (as in [12]). Furthermore, for some p ≥ 2 if d = 2 and p ∈ [2, 6] if d = 3, the following requirements hold

(i) there exist constants β

i

≥ 0 (i = 1, . . . , 10) such that for any v ∈ IR

|I

₀

(.; v)| ≤ β

₁

+ β

₂

| v |

^p−1

,

|I

₁

(.; v)| ≤ β

₃

+ β

₄

|v|

^p/2−1

,

|I

₂

(.; v)| ≤ β

₅

+ β

₆

|v|

^p/2

,

|E

g

(.; v)| ≤ β

₇

+ β

₈

|v|

^p

, |g

₂

(.; v)| ≤ β

₉

+ β

₁₀

|v|

^p−2

,

(2.4)

where E

g

is the primitive of g

1

.

(ii) there exist constants µ

₁

> 0, µ

₂

> 0, µ

i

≥ 0 (for i = 3, 7) such that for any (v, w) ∈ IR

²

: µ

1

v I(.; v, w) + w G(.; v, w) ≥ µ

2

| v |

^p

− µ

3

µ

1

| v |

²

+ | w |

²

− µ

4

,

E

g

(., v) ≥ µ

₅

| v |

^p

− µ

₆

| v |

²

− µ

₇

. (2.5) In order to assure the uniqueness of solution we assume that

(H3) The Nemytskii operators I and G satisfy Carath´eodory conditions and there exists some µ > 0 such the operator F

µ

: IR

²

→ IR

²

defined by

F

µ

(.; v) = µ(I(.; v)) G (.; v)

!

, ∀v = (v, w) ∈ IR

²

, (2.6)

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satisfies a one-sided Lipschitz condition (see e.g., [14, 70]): there exists a constant C

L

> 0 such that (∀v

i

= (v

i

, w

i

) ∈ IR

²

, i = 1, 2)

F

µ

(.; v

₁

) − F

µ

(.; v

₂

)

· (v

₁

− v

₂

) ≥ −C

L

kv

₁

− v

₂

k

²

. (2.7) Lemma 2.2. ( [9]) Assume that F

µ

is di ff erentiable with respect to (φ, u) and denote by λ

1

(φ, u) ≤ λ

₂

(φ, u) the eigenvalues of symmetrical part of Jacobian matrix ∇F

µ

(φ, u):

Q

µ

(φ, u) = 1 2

∇F

µ

(φ, u)

^T

+ ∇F

µ

(φ, u) . If there exists a constant C

F

independent of φ and u such as:

C

F

≤ λ

₁

(φ, u) ≤ λ

₂

(φ, u), (2.8)

then F

µ

satisfies the hypothesis (H3).

Lemma 2.3. ( [9]) Let assumptions (2.3), (H1) and (H2)

_p

be fulfilled. For (φ, u) ∈ L

^p

( Ω ) × H and a.e., t, there exist constants C

i

> 0 (i = 1, 6) such that

kI(., t; φ, u)k

_Lp0

(ΩH)

≤ C

₁

+ C

₂

kφk

_L^p/pp(Ω⁰H)

+ C

₃

kuk

^2/p_H ⁰

H

,

kG(., t; φ, u)k

L²(ΩH)

≤ C

₄

+ C

₅

kφk

^p/2_Lp(ΩH)

+ C

₆

kuk

_H_H

, (2.9) where p

⁰

is such that

¹_p

+

_p¹⁰

= 1.

In this work we assume that p = 4. The considered functions I

_i

, in this paper, include the three classical type models in which assumptions (H1), (H2)

₄

and (H3) are satisfied namely the Rogers-McCulloch [57] (RM), Fitz-Hugh-Nagumo [39] (FHN) and Aliev-Panfilov [61](LAP) models as follows. The function I

₀

is defined by a cubic reaction term of the form I

₀

(.; v) = b

₁

(.)v(v − r)(v − 1), and the functions I

₁

and I

₂

are given by

(a) for RM type model : I

₁

(.; v) = b

2

(.)v, I

₂

(; , v) = − b

3

(.)v, (b) for FHN type model : I

₁

(.; v) = b

2

(.), I

₂

(; , v) = − b

3

(.)v,

(c) for LAP type model : I

₁

(.; v) = b

2

(.)v, I

₂

(; , v) = b

3

(.)v(r + 1 − v),

where b

i

∈ W

^1,∞

(Q), i = 1, 3, are su ffi ciently regular functions from Q into IR

⁺^,∗

and r ∈ [0, 1]. We obtain easily the following Lemma.

Lemma 2.4. The following properties hold. For all v

1

, v

2

in IR we have I

₀

(.; v

1

) − I

₀

(.; v

2

) = b

1

(v

1

− v

2

)

v

²₁

+ v

²₂

+ v

1

v

2

− (r + 1)(v

1

+ v

2

) + r and

(a) for RM type model : I

₁

(.; v

1

) − I

₁

(.; v

2

) = b

2

(v

1

− v

2

), I

₂

(.; v

1

) − I

₂

(.; v

2

) = − b

3

(v

1

− v

2

), (b) for FHN type model : I

₁

(.; v

1

) − I

₁

(.; v

2

) = 0, I

₂

(.; v

1

) − I

₂

(.; v

2

) = − b

3

(v

1

− v

2

),

(c) for LAP type model : I

₁

(.; v

1

) − I

₁

(.; v

2

) = b

2

(v

₁

− v

2

),

I

₂

(.; v

₁

) − I

₂

(.; v

₂

) = b

₃

(v

₁

− v

₂

)((r + 1) − v

₁

− v

₂

).

For the sake of simplicity, we shall write I

_i

(ψ), I(ψ, v) and G(ψ, v) in place of I

_i

(x, t; ψ), I(x, t; ψ, v)

and G(x, t; ψ, v), respectively (for i = 0, 2).

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3. Fractional calculus and a generalized Gronwall’s inequality

Fractional di ff erential equations have been studied by many investigations in recent years and the idea of defining a derivative of fractional order (non-integer order) dates back to Leibnitz [49]. Since 19th century, di ff erent authors have considered this problem e.g., Riemann, Liouville, Hadmard, Hardy, Littlewood and Caputo among others. Fractional integrals and derivatives have proved to be useful in real applications, since they arise naturally in many biological phenomena such as viscoelasticity, neurobiology and chaotic systems, see for instance [3, 34, 36, 55, 62, 76, 83].

The classical and the most used form of fractional calculus is given by the Riemann-Liouville and Caputo derivatives. In contrast to this nonlocal Riemann-Liouville derivative operator, when solving di ff erential equations, is the use of Caputo fractional derivative [22] in which it is not necessary to define the fractional order initial conditions.

3.1. Definitions and basic results

The object of this section is to give a brief introduction to some definitions and basic results in fractional calculus in the Riemann-Liouville sense and Caputo sense. Let γ ∈]0, 1] and X be a Banach space, we start from a formal level and assume the given functions f : t ∈ (a, T ) → f (t) ∈ X and g : t ∈ (a, T ) → f (t) ∈ X are su ffi ciently smooth (with −∞ < a < T < ∞).

Definition 3.1. The forward and backward Riemann-Liouville fractional integrals of fractional order γ on (a, T ) are defined, respectively, by (t ∈ (a, T ))

I

_a^γ₊

f

(t) = 1 Γ (γ)

Z

t

a

(t − τ)

^γ−1

f (τ)dτ, I

_T^γ−

f

(t) = 1 Γ (γ)

Z

T

t

(τ − t)

^γ−1

f (τ)dτ,

(3.1)

where Γ (z) = Z

∞ 0

e

^τ

τ

^z−1

dτ is the Euler Γ-function.

The basic equality for the fractional integral is (from Fubini’s Theorem and the relationship between Γ -function and β-function)

I

_a^γ₊¹

h I

_a^γ₊²

f i

= I

^γ_a₊¹⁺^γ²

f

(3.2) and holds for a L

^p

-function f (1 ≤ p ≤ ∞).

Definition 3.2. The forward Riemann-Liouville and Caputo derivatives of fractional order γ on (a, T ) are defined, respectively, by (t ∈ (a, T ))

D

^γ_a₊

f

(t) = d dt (I

_a^1−γ₊

f

(t)) = 1 Γ (1 − γ)

d dt (

Z

t

a

(t − τ)

^−γ

f (τ)dτ),

∂

^γ_a+

f

(t) = I

_a^1−γ+

"

d f (t) dt

#

= 1

Γ (1 − γ) Z

t

a

(t − τ)

^−γ

d f dt (τ)dτ.

(3.3)

From (3.2) we can deduce the following relation between fractional integral and Caputo derivative f (t) = f (a) + I

_a^γ+

h ∂

^γ_a₊

f i

(t) = f (a) + 1 Γ (γ)

Z

t

a

(t − τ)

^γ−1

∂

^γ_a₊

f (τ)dτ. (3.4)

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Definition 3.3. The backward Riemann-Liouville and Caputo derivatives of fractional order γ, on (a, T ) are defined, respectively, by (t ∈ (a, T ))

D

^γ_T−

f (t) = − d dt (I

_T^1−γ−

f

(t)) = − 1 Γ (1 − γ)

d dt (

Z

T

t

(τ − t)

^−γ

f (τ)dτ),

∂

^γ_T−

f

(t) = −I

_T^1−γ−

"

d f (t) dt

#

= − 1 Γ (1 − γ)

Z

T

t

(τ − t)

^−γ

d f dt (τ)dτ.

(3.5)

By substitution a → −∞, the following definition is obtained.

Definition 3.4. The Liouville-Weyl fractional integral and the Caputo fractional derivative on the real axis are defined, respectively, by (t ∈ (−∞, T ))

I

_−∞^γ

f

(t) = 1 Γ (γ)

Z

t

−∞

(t − τ)

^γ−¹

f (τ)dτ,

∂

^γ_−∞

f (t) = I

_−∞^1−γ

"

d f dt (t)

#

= 1

Γ (1 − γ) Z

t

−∞

(t − τ)

^−γ

d f dt (τ)dτ.

(3.6)

Remark 3.1. 1. For γ −→ 1 the forward (resp. backward) Riemann-Liouville and Caputo derivatives of fractional order γ of f converge to the classical derivative

^{d f}_dt

(resp. to −

^{d f}

dt

).

Moreover, Riemann-Liouville fractional derivative of fractional order γ of constant function t −→ f (t) = k is not 0 since D

^γ_a₊

f (t) =

_Γ_(1−γ)^k _dt^d

( R

t

a

(t − τ)

^−γ

dτ) =

^k(t−a)_Γ_(1−γ)^−γ

.

2. It is possible to show that the di ff erence between Riemann-Liouville and Caputo fractional derivatives depends only on the values of f on endpoints a and T . More precisely, for

f ∈ C

¹

([a, T ], X) we have

D

^γ_a₊

f (t) = ∂

^γ_a₊

f (t) +

^f^(a)(t−a)_Γ_(1−γ)^−γ

,

D

^γ_T−

f (t) = ∂

^γ_T−

f (t) +

^f^(T)(T−t)_Γ_(1−γ)^−γ

. (3.7) From [42], we have the following results

Lemma 3.1. (Continuity properties of fractional integral in L

^p

spaces on (a, T )) The fractional integral I

_a^γ+

is a continuous operator from

(i) L

^p

(a, T ) into L

^p

(a, T ), for any p ≥ 1,

(ii) L

^p

(a, T ) into L

^r

(0, T ), for any p ∈ (1, 1/γ) and r ∈ [1, p/(1 − γ p)], (iii) L

^p

(a, T ) into C

^0,γ−1/p

([a, T ]), for any p ∈]1/γ, + ∞[,

(iii) L

^1/γ

(a, T ) into L

^p

(a, T ), for any p ∈ [1, + ∞) (iv) L

^∞

(a, T ) into C

^0,γ

([a, T ]).

From Lemma 3.1 and (3.4) we can deduce the following corollary.

Lemma 3.2. Let X be a Banach space and γ ∈]0, 1]. Suppose the Caputo derivative ∂

^γ_a₊

f ∈ L

^p

(a, T ; X) and p >

¹_γ

, then f ∈ C

^0,γ−1/p

([a, T ]; X).

We also need for our purposes the fractional integration by parts in the formulas (see for instance

[2, 47, 86])

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Lemma 3.3. Let 0 < γ ≤ 1 and p, q ≥ 1 with 1/p + 1/q ≤ 1 + γ. Then (i) if g is a L

^p

-function on (a, T ) and g is a L

^q

-function on (a, T ), then

Z

T

a

( f (τ), I

_a^γ+

g (τ))

X

dτ = Z

T a

(g(τ), I

_T^γ−

f (τ))

X

dτ, (ii) if f ∈ I

_T^γ−

(L

^p

) and g ∈ I

_a^γ₊

(L

^q

), then

Z

T

a

( f (τ), D

^γ_a₊

g(τ))

X

dτ = Z

T

a

(g(τ), D

^γ_T−

f (τ))

X

dτ.

Lemma 3.4. Let 0 < γ ≤ 1, and g be L

^p

-function on (a, T ) (for p ≥ 1) and f be absolutely continuous function on [a, T ]. Then

(i) Z

T

a

(∂

^γ_a₊

f (τ), g(τ))

X

dτ = − Z

T

a

(D

^γ_T−

g(τ), f (τ))

X

dτ + |(I

_T^1−γ−

g (τ), f (τ))

X

|

^T_a

, (ii)

Z

T

a

(D

^γ_a+

f (τ), g(τ))

X

dτ = − Z

T

a

(D

^γ_T−

g(τ), f (τ))

X

dτ + (I

_T^1−γ−

g

(T

⁻

), f (T ))

X

, (iii)

Z

T

a

(D

^γ_T−

f (τ), g(τ))

X

dτ = − Z

T

a

(D

^γ_a₊

g(τ), f (τ))

X

dτ − (I

_a^1−γ₊

g

(a

⁺

), f (a))

X

. From [85], we can deduce the following Lemma.

Lemma 3.5. (A generalized Gronwall’s inequality)

Assume γ > 0, h is a nonnegative function locally integrable on (0, T ) and b is a nonnegative, bounded, nondecreasing continuous function defined on [0, T ). Let f be a nonnegative and locally integrable function on (0, T ) with

f (t) ≤ h(t) + b(t)I

₀^γ₊

[ f ](t).

Then (for t ∈ (0, T ))

f (t) ≤ h(t) + Z

t

0

∞

X

k=1

h(τ)(t − τ)

^kγ−1

(b(t))

^k

Γ (kγ) dτ.

If in addition h is a nondecreasing function on (0, T ), then f (t) ≤ h(t)E

_γ,1

(b(t)t

^γ

).

The used function E

θ1,θ2

is the classical two-parametric Mittag-Le ffl er function (usually denoted by E

_θ₁

if θ

₂

= 1) which is defined by

E

_θ₁_,θ₂

(z) =

∞

X

k=0

1 Γ (kθ

1

+ θ

2

) z

^k

. (3.8)

The function E

_θ₁_,θ₂

is an entire function of the variable z for any θ

₁

, θ

₂

∈ C, l Re(θ

₁

) > 0.

Next we declare a compactness theorem in Hilbert spaces. Assume that X

₀

, X

₁

and X are Hilbert spaces with

X

₀

, → X , → X

₁

being continuous and X

₀

, → X is compact. (3.9) The Fourier transform of f : IR → X

₁

is defined by b f (τ) = Z

∞

−∞

exp(−2iπ sτ) f (s)ds and we have

Lemma (see e.g., [68])

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Lemma 3.6. Let γ ∈ (0, 1) and f be a L

¹

-function in IR with compact support. Then (i) I d

_−∞^γ

f (τ) = (2iπτ)

^−γ

b f (τ),

(ii) ∂ [

^γ_−∞

f (τ) = (2iπτ)

^γ

b f (τ).

We define the Hilbert space W

^γ

(IR; X

₀

, X

₁

), for a given γ > 0, by

W

^γ

(IR; X

0

, X

₁

) = {v ∈ L

²

(IR, X

₀

) | ∂

^γ_−∞

f ∈ L

²

(IR, X

₁

)}, endowed with the norm

k v k

_W^γ

= k v k

_L2

IR

^,X0

+ k| τ |

^γ

b v k

_L2

IR

^,X1

1/2

. For any subset K of IR, we define the subspace W

^γK

of W

^γ

by

W

^γ_K

(IR; X

₀

, X

₁

) = {v ∈ W

^γ

(IR; X

₀

, X

₁

) | support of v ⊂ K}.

From Lemma 3.6 and similar arguments to drive Theorem 2.2 in [77], we have the following compactness result.

Theorem 3.1. Let X

₀

, X

₁

and X be Hilbert spaces with the injection (3.9). Then for any bounded set K and any γ > 0, the injection of W

^γ_K

(IR; X

₀

, X

₁

) into L

²

(IR; X) is compact.

3.2. A generalized Gronwall inequality to a coupled fractional di ff erential equation

In this section we present a new generalized Gronwall inequality with singularity in the context of a coupled fractional di ff erential equations.

Theorem 3.2. Assume that γ

₁

and γ

₂

be in ]0,1] with γ

₂

≤ γ

₁

and Ψ , h

i

, for i = 1, 2, are nonnegative functions locally integrable on (0, T ). If f and g are nonnegative functions locally integrable, with

f (0) = f

0

, g(0) = g

0

, and satisfy the inequalities (for t ∈ (0, T ))

∂

^γ₀₊¹

f (t) ≤ h

₁

(t) + c

₁

( f (t) + g(t)) + η Ψ (t),

∂

^γ₀+²

g(t) + ξ Ψ (t) ≤ h

2

(t) + c

2

( f (t) + g(t)), (3.10) where c

i

> 0, i = 1, 2, η ≥ 0 and ξ ≥ 0 are four constants with η ≤ ξ Γ (γ

1

)

Γ (γ

₂

)T

^(γ¹^−γ²⁾

. Then we have the following estimate

F(t) ≤ F

₀

E

_γ₂_,1

(d

₁

t

^γ²

) +

∞

X

k=0

d

₁^k

I

₀^kγ₊²⁺^γ¹

[h

₁

] (t) +

∞

X

k=0

d

^k₁

I

₀^(k₊^+1)γ²

[h

₂

] (t), (3.11)

where F = f + g, F

₀

= f

₀

+ g

₀

, d

₀

= T

^(γ¹^−γ²⁾

Γ (γ

₂

)

Γ (γ

1

) and d

₁

= c

₁

d

₀

+ c

₂

.

The function E

.,.

is the Mittag-Le ffl er function which is defined by (3.8).

(17)

Proof. From (3.4) we can deduce that Γ (γ

₁

)( f (t) − f

₀

) ≤

Z

t

0

(t − τ)

^γ¹⁻¹

h

₁

(τ)dτ + c

₁

Z

t

0

(t − τ)

^γ¹⁻¹

( f (τ) + g(τ))dτ + η Z

t

0

(t − τ)

^γ¹⁻¹

Ψ (τ)dτ, Γ (γ

2

)(g(t) − g

0

) ≤

Z

t

0

(t − τ)

^γ²⁻¹

h

2

(τ)dτ + c

2

Z

t

0

(t − τ)

^γ²⁻¹

( f (τ) + g(τ))dτ

−ξ Z

t 0

(t − τ)

^γ²⁻¹

Ψ (τ)dτ.

(3.12)

Put F = f + g, then

( Γ (γ

₁

) Γ (γ

₂

))(F(t) − F

₀

) ≤ Z

t

0

( Γ (γ

₂

)(t − τ)

^γ¹⁻¹

h

₁

(τ) + Γ (γ

₁

)(t − τ)

^γ²⁻¹

h

₂

(τ))dτ + Z

t

0

η Γ (γ

₂

)(t − τ)

^γ¹⁻¹

− ξ Γ (γ

₁

)(t − τ)

^γ²⁻¹

Ψ (τ)dτ +

Z

t

0

c

1

Γ (γ

2

)(t − τ)

^γ¹⁻¹

+ c

2

Γ (γ

1

)(t − τ)

^γ²⁻¹

F(τ)dτ.

(3.13)

We can deduce that (since t ∈ (0, T ) and γ

₁

− γ

₂

≥ 0) ( Γ (γ

₁

) Γ (γ

₂

))(F(t) − F

₀

) ≤ Γ (γ

₂

)

Z

t

0

(t − τ)

^γ¹⁻¹

h

₁

(τ)dτ + Γ (γ

₁

) Z

t

0

(t − τ)

^γ²⁻¹

h

₂

(τ))dτ + (η Γ (γ

2

)T

^(γ¹^−γ²⁾

− ξ Γ (γ

1

))

Z

t

0

(t − τ)

^γ²⁻¹

Ψ (τ)dτ + (c

₁

Γ (γ

₂

)T

^(γ¹^−γ²⁾

+ c

₂

Γ (γ

₁

))

Z

t

0

(t − τ)

^γ²⁻¹

F(τ)dτ.

(3.14)

Since η Γ (γ

2

)T

^(γ¹^−γ²⁾

− ξ Γ (γ

1

) ≤ 0, then ( Γ (γ

₁

) Γ (γ

₂

))(F(t) − F

₀

) ≤ Γ (γ

₂

)

Z

t

0

(t − τ)

^γ¹⁻¹

h

₁

(τ)dτ + Γ (γ

₁

) Z

t

0

(t − τ)

^γ²⁻¹

h

₂

(τ))dτ + (c

₁

Γ (γ

2

)T

^(γ¹^−γ²⁾

+ c

2

Γ (γ

1

))

Z

t

0

(t − τ)

^γ²⁻¹

F(τ)dτ.

(3.15)

Thus

F(t) ≤ F

₀

+ I

₀^γ₊¹

[h

₁

] (t) + I

₀^γ₊²

[h

₂

] (t) + d

₁

I

₀^γ₊²

[F] (t), (3.16) where d

0

= T

^(γ¹^−γ²⁾

Γ (γ

2

)

Γ (γ

₁

) and d

1

= c

1

Γ (γ

2

)T

^(γ¹^−γ²⁾

+ c

2

Γ (γ

1

)

Γ (γ

₁

) = c

1

d

0

+ c

2

.

Finally, from Lemma 3.5 and relation (3.2), we get (since I

₀^kγ₊²

[F] (0) = F

₀

I

^kγ₀₊²

[.1] =

_Γ^F₍₁⁰₊^t^kγ_kγ²₂₎

) F(t) ≤ F

₀

∞

X

k=0

1 Γ (1 + kγ

₂

) (d

₁

t

^γ²

)

^k

+

∞

X

k=0

d

₁^k

I

₀^kγ₊²⁺^γ¹

[h

₁

] (t) +

∞

X

k=0

d

^k₁

I

₀^(k₊⁺^1)γ²

[h

₂

] (t).

This completes the proof.